Third Chern class of a sheaf on a threefold Let $X$ be a smooth projective threefold. Let $D$ be a smooth irreducible divisor on $X$ and $Z$ be an irreducible smooth divisor on $D$. We have inclusions: $$ j:Z\hookrightarrow D\text{ and } i :D\hookrightarrow X.$$ Consider the sheaf $M=i_*j_*O_Z$ on $X$. I am trying to compute the Chern classes of $M$. I have obtained the following: $c_1(M)=0$ since $Z$ is of codimension 2 in $X$. $c_2(M)=-[Z]$. Is this correct? What about $c_3$. Is it zero? I am not able to justify that. Also if $L$ is a line bundle on $D$, what are the Chern classes of $N=i_*L$. I know that $c_1(N)=[D]$. What are $c_2$ and $c_3$? Thanks in advance!

If $Z=E\cap D$, wuth $E,D$ divisors on $X$, we have an exact sequence, $0\to O_X(-D-E)\to O_X(-D)\oplus O_X(-E)\to O_X\to O_Z\to 0$. So, Whitney sum formula says, $c(O_Z)c(O_X(-D)\oplus c(O_X(-E))=c(O_X)c(O_X(-D-E))$, where $c$ is the total chern class. If we denote by $a=c_1(O_X(D))$, $a_2=c_1(O_X(E))$, we get, $c(O_Z)(1-a)(1-b)=1-(a+b)$. From this, you can calculate all the chern classes of $O_Z$ easily by elementary algebra. We have $c(O_Z)=(1-(a+b))(1-a)^{-1}(1-b)^{-1}= (1-(a+b))(1+a+a^2+a^3)(1+b+b+b^2+b^3)$. Thus $c_1(O_Z)=0$ as you said. $c_2=a^2+ab+b^2-(a+b)^2=-ab$, again, this is the class of $-[Z]$ as you said. Finally, $c_3=a^3+a^2b+ab^2+b^3-(a+b)(a^2+ab+b^2)=-ba^2-ab^2=-(a+b)ab$.